sqrtexp (problem 3.4.4)

Average Error: 40.4 → 0.4

Time: 8.5s

Precision: binary64

Cost: 66244

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{if}\;t_0 \leq 1.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left({x}^{2} \cdot \left(\sqrt{2} \cdot 0.25 + \sqrt{8} \cdot -0.03125\right) + \sqrt{2} \cdot \left({x}^{3} \cdot 0.036458333333333336\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0)))))
   (if (<= t_0 1.2)
     t_0
     (+
      (sqrt 2.0)
      (*
       0.5
       (+
        (/ x (sqrt 2.0))
        (+
         (* (pow x 2.0) (+ (* (sqrt 2.0) 0.25) (* (sqrt 8.0) -0.03125)))
         (* (sqrt 2.0) (* (pow x 3.0) 0.036458333333333336)))))))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

↓

double code(double x) {
	double t_0 = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
	double tmp;
	if (t_0 <= 1.2) {
		tmp = t_0;
	} else {
		tmp = sqrt(2.0) + (0.5 * ((x / sqrt(2.0)) + ((pow(x, 2.0) * ((sqrt(2.0) * 0.25) + (sqrt(8.0) * -0.03125))) + (sqrt(2.0) * (pow(x, 3.0) * 0.036458333333333336)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
    if (t_0 <= 1.2d0) then
        tmp = t_0
    else
        tmp = sqrt(2.0d0) + (0.5d0 * ((x / sqrt(2.0d0)) + (((x ** 2.0d0) * ((sqrt(2.0d0) * 0.25d0) + (sqrt(8.0d0) * (-0.03125d0)))) + (sqrt(2.0d0) * ((x ** 3.0d0) * 0.036458333333333336d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}

↓

public static double code(double x) {
	double t_0 = Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
	double tmp;
	if (t_0 <= 1.2) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(2.0) + (0.5 * ((x / Math.sqrt(2.0)) + ((Math.pow(x, 2.0) * ((Math.sqrt(2.0) * 0.25) + (Math.sqrt(8.0) * -0.03125))) + (Math.sqrt(2.0) * (Math.pow(x, 3.0) * 0.036458333333333336)))));
	}
	return tmp;
}

Python

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))

↓

def code(x):
	t_0 = math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))
	tmp = 0
	if t_0 <= 1.2:
		tmp = t_0
	else:
		tmp = math.sqrt(2.0) + (0.5 * ((x / math.sqrt(2.0)) + ((math.pow(x, 2.0) * ((math.sqrt(2.0) * 0.25) + (math.sqrt(8.0) * -0.03125))) + (math.sqrt(2.0) * (math.pow(x, 3.0) * 0.036458333333333336)))))
	return tmp

Julia

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end

↓

function code(x)
	t_0 = sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
	tmp = 0.0
	if (t_0 <= 1.2)
		tmp = t_0;
	else
		tmp = Float64(sqrt(2.0) + Float64(0.5 * Float64(Float64(x / sqrt(2.0)) + Float64(Float64((x ^ 2.0) * Float64(Float64(sqrt(2.0) * 0.25) + Float64(sqrt(8.0) * -0.03125))) + Float64(sqrt(2.0) * Float64((x ^ 3.0) * 0.036458333333333336))))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
end

↓

function tmp_2 = code(x)
	t_0 = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
	tmp = 0.0;
	if (t_0 <= 1.2)
		tmp = t_0;
	else
		tmp = sqrt(2.0) + (0.5 * ((x / sqrt(2.0)) + (((x ^ 2.0) * ((sqrt(2.0) * 0.25) + (sqrt(8.0) * -0.03125))) + (sqrt(2.0) * ((x ^ 3.0) * 0.036458333333333336)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1.2], t$95$0, N[(N[Sqrt[2.0], $MachinePrecision] + N[(0.5 * N[(N[(x / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * 0.25), $MachinePrecision] + N[(N[Sqrt[8.0], $MachinePrecision] * -0.03125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[x, 3.0], $MachinePrecision] * 0.036458333333333336), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (/.f64 (-.f64 (exp.f64 (*.f64 2 x)) 1) (-.f64 (exp.f64 x) 1))) < 1.19999999999999996

   1. Initial program 0.0

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

   if 1.19999999999999996 < (sqrt.f64 (/.f64 (-.f64 (exp.f64 (*.f64 2 x)) 1) (-.f64 (exp.f64 x) 1)))

   1. Initial program 61.1

      \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

   2. Taylor expanded in x around 0 0.6

      \[\leadsto \color{blue}{\sqrt{2} + \left(0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}} + \left(0.5 \cdot \frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right)} \]

   3. Simplified 0.6

      \[\leadsto \color{blue}{\sqrt{2} + 0.5 \cdot \left({x}^{3} \cdot \frac{0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{2}}{\sqrt{2}} + \left(\frac{x}{\sqrt{2}} + {x}^{2} \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{\sqrt{2}}\right)\right)} \]
      Proof

      [Start] 0.6	\[ \sqrt{2} + \left(0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}} + \left(0.5 \cdot \frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) \]
      rational.json-simplify-1 [=>] 0.6	\[ \sqrt{2} + \color{blue}{\left(\left(0.5 \cdot \frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + 0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}}\right)} \]
      rational.json-simplify-1 [=>] 0.6	\[ \sqrt{2} + \left(\color{blue}{\left(0.5 \cdot \frac{x}{\sqrt{2}} + 0.5 \cdot \frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}}\right)} + 0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}}\right) \]
      rational.json-simplify-2 [=>] 0.6	\[ \sqrt{2} + \left(\left(0.5 \cdot \frac{x}{\sqrt{2}} + \color{blue}{\frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} \cdot 0.5}\right) + 0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}}\right) \]
      rational.json-simplify-51 [=>] 0.6	\[ \sqrt{2} + \left(\color{blue}{0.5 \cdot \left(\frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} + \frac{x}{\sqrt{2}}\right)} + 0.5 \cdot \frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}}\right) \]
      rational.json-simplify-2 [=>] 0.6	\[ \sqrt{2} + \left(0.5 \cdot \left(\frac{\left(0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}\right) \cdot {x}^{2}}{\sqrt{2}} + \frac{x}{\sqrt{2}}\right) + \color{blue}{\frac{\left(0.16666666666666666 - 0.5 \cdot \frac{0.5 - {\left(\frac{0.5}{\sqrt{2}}\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}\right) \cdot {x}^{3}}{\sqrt{2}} \cdot 0.5}\right) \]

   4. Taylor expanded in x around -inf 0.6

      \[\leadsto \sqrt{2} + 0.5 \cdot \color{blue}{\left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{3}}\right) \cdot {x}^{2} + \left(\frac{\left(0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right) \cdot {x}^{3}}{\sqrt{2}} + \frac{x}{\sqrt{2}}\right)\right)} \]

   5. Simplified 0.6

      \[\leadsto \sqrt{2} + 0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{2}} + \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{\sqrt{8}}\right) \cdot {x}^{2} + {x}^{3} \cdot \frac{0.07291666666666667}{\sqrt{2}}\right)\right)} \]
      Proof

      [Start] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{3}}\right) \cdot {x}^{2} + \left(\frac{\left(0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right) \cdot {x}^{3}}{\sqrt{2}} + \frac{x}{\sqrt{2}}\right)\right) \]
      rational.json-simplify-41 [<=] 0.6	\[ \sqrt{2} + 0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{2}} + \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{3}}\right) \cdot {x}^{2} + \frac{\left(0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right) \cdot {x}^{3}}{\sqrt{2}}\right)\right)} \]
      exponential.json-simplify-24 [=>] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{\color{blue}{\sqrt{{2}^{3}}}}\right) \cdot {x}^{2} + \frac{\left(0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right) \cdot {x}^{3}}{\sqrt{2}}\right)\right) \]
      metadata-eval [=>] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{\sqrt{\color{blue}{8}}}\right) \cdot {x}^{2} + \frac{\left(0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right) \cdot {x}^{3}}{\sqrt{2}}\right)\right) \]
      rational.json-simplify-49 [=>] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left(\left(0.5 \cdot \frac{1}{\sqrt{2}} - 0.25 \cdot \frac{1}{\sqrt{8}}\right) \cdot {x}^{2} + \color{blue}{{x}^{3} \cdot \frac{0.16666666666666666 - 0.25 \cdot \left(0.5 - 0.25 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}}}\right)\right) \]

   6. Applied egg-rr 0.6

      \[\leadsto \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \color{blue}{\left(\left(\left(-0.25 \cdot \left(\sqrt{8} \cdot 0.125 - \sqrt{2}\right)\right) \cdot {x}^{2} + {x}^{3} \cdot \left(\sqrt{2} \cdot 0.036458333333333336\right)\right) - 0\right)}\right) \]

   7. Simplified 0.6

      \[\leadsto \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \color{blue}{\left({x}^{2} \cdot \left(\sqrt{2} \cdot 0.25 + \sqrt{8} \cdot -0.03125\right) + \sqrt{2} \cdot \left({x}^{3} \cdot 0.036458333333333336\right)\right)}\right) \]
      Proof

      [Start] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left(\left(\left(-0.25 \cdot \left(\sqrt{8} \cdot 0.125 - \sqrt{2}\right)\right) \cdot {x}^{2} + {x}^{3} \cdot \left(\sqrt{2} \cdot 0.036458333333333336\right)\right) - 0\right)\right) \]
      rational.json-simplify-5 [=>] 0.6	\[ \sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \color{blue}{\left(\left(-0.25 \cdot \left(\sqrt{8} \cdot 0.125 - \sqrt{2}\right)\right) \cdot {x}^{2} + {x}^{3} \cdot \left(\sqrt{2} \cdot 0.036458333333333336\right)\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leq 1.2:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + 0.5 \cdot \left(\frac{x}{\sqrt{2}} + \left({x}^{2} \cdot \left(\sqrt{2} \cdot 0.25 + \sqrt{8} \cdot -0.03125\right) + \sqrt{2} \cdot \left({x}^{3} \cdot 0.036458333333333336\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	39620

\[\begin{array}{l} t_0 := \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2}\\ \end{array} \]

Alternative 2
Error	18.3
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))